Finite element methods for multicomponent convection-diffusion

We develop finite element methods for coupling the steady-state Onsager--Stefan--Maxwell equations to compressible Stokes flow. These equations describe multicomponent flow at low Reynolds number, where a mixture of different chemical species within a common thermodynamic phase is transported by convection and molecular diffusion. Developing a variational formulation for discretizing these equations is challenging: the formulation must balance physical relevance of the variables and boundary data, regularity assumptions, tractability of the analysis, enforcement of thermodynamic constraints, ease of discretization, and extensibility to the transient, anisothermal, and non-ideal settings. To resolve these competing goals, we employ two augmentations: the first enforces the mass-average constraint in the Onsager--Stefan--Maxwell equations, while its dual modifies the Stokes momentum equation to enforce symmetry. Remarkably, with these augmentations we achieve a Picard linearization of symmetric saddle point type, despite the equations not possessing a Lagrangian structure. Exploiting the structure of linear irreversible thermodynamics, we prove the inf-sup condition for this linearization, and identify finite element function spaces that automatically inherit well-posedness. We verify our error estimates with a numerical example, and illustrate the application of the method to non-ideal fluids with a simulation of the microfluidic mixing of hydrocarbons

Variational Problems with Fractional Derivatives: Euler-Lagrange Equations

We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Also, for the standard case when these two bounds coincide, we derive a new form of Euler-Lagrange equations. We use approximations for fractional derivatives in the Lagrangian and obtain the Euler-Lagrange equations which approximate the initial Euler-Lagrange equations in a weak sense

Fractional variational calculus of variable order

We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann-Liouville while derivatives are of Caputo type.Comment: Submitted 26-Sept-2011; accepted 18-Oct-2011; withdrawn by the authors 21-Dec-2011; resubmitted 27-Dec-2011; revised 20-March-2012; accepted 13-April-2012; to 'Advances in Harmonic Analysis and Operator Theory', The Stefan Samko Anniversary Volume (Eds: A. Almeida, L. Castro, F.-O. Speck), Operator Theory: Advances and Applications, Birkh\"auser Verlag (http://www.springer.com/series/4850

Progress in Classical and Quantum Variational Principles

We review the development and practical uses of a generalized Maupertuis least action principle in classical mechanics, in which the action is varied under the constraint of fixed mean energy for the trial trajectory. The original Maupertuis (Euler-Lagrange) principle constrains the energy at every point along the trajectory. The generalized Maupertuis principle is equivalent to Hamilton's principle. Reciprocal principles are also derived for both the generalized Maupertuis and the Hamilton principles. The Reciprocal Maupertuis Principle is the classical limit of Schr\"{o}dinger's variational principle of wave mechanics, and is also very useful to solve practical problems in both classical and semiclassical mechanics, in complete analogy with the quantum Rayleigh-Ritz method. Classical, semiclassical and quantum variational calculations are carried out for a number of systems, and the results are compared. Pedagogical as well as research problems are used as examples, which include nonconservative as well as relativistic systems

The Gaseous Electronics Conference radio‐frequency reference cell: A defined parallel‐plate radio‐frequency system for experimental and theoretical studies of plasma‐processing discharges

A ‘‘reference cell’’ for generating radio‐frequency (rf) glow discharges in gases at a frequency of 13.56 MHz is described. The reference cell provides an experimental platform for comparing plasma measurements carried out in a common reactor geometry by different experimental groups, thereby enhancing the transfer of knowledge and insight gained in rf discharge studies. The results of performing ostensibly identical measurements on six of these cells in five different laboratories are analyzed and discussed. Measurements were made of plasma voltage and current characteristics for discharges in pure argon at specified values of applied voltages, gas pressures, and gas flow rates. Data are presented on relevant electrical quantities derived from Fourier analysis of the voltage and current wave forms. Amplitudes, phase shifts, self‐bias voltages, and power dissipation were measured. Each of the cells was characterized in terms of its measured internal reactive components. Comparing results from different cells provides an indication of the degree of precision needed to define the electrical configuration and operating parameters in order to achieve identical performance at various laboratories. The results show, for example, that the external circuit, including the reactive components of the rf power source, can significantly influence the discharge. Results obtained in reference cells with identical rf power sources demonstrate that considerable progress has been made in developing a phenomenological understanding of the conditions needed to obtain reproducible discharge conditions in independent reference cells.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70394/2/RSINAK-65-1-140-1.pd

Identifying predictors of attitudes towards local onshore wind development with reference to an English case study

The threats posed by climate change are placing governments under increasing pressure to meet electricity demand from low-carbon sources. In many countries, including the UK, legislation is in place to ensure the continued expansion of renewable energy capacity. Onshore wind turbines are expected to play a key role in achieving these aims. However, despite high levels of public support for onshore wind development in principle, specific projects often experience local opposition. Traditionally this difference in general and specific attitudes has been attributed to NIMBYism (not in my back yard), but evidence is increasingly calling this assumption into question. This study used multiple regression analysis to identify what factors might predict attitudes towards mooted wind development in Sheffield, England. We report on the attitudes of two groups; one group (target) living close to four sites earmarked for development and an unaffected comparison group (comparison). We found little evidence of NIMBYism amongst members of the target group; instead, differences between general and specific attitudes appeared attributable to uncertainty regarding the proposals. The results are discussed with respect to literature highlighting the importance of early, continued and responsive community involvement in combating local opposition and facilitating the deployment of onshore wind turbines. (C) 2009 Elsevier Ltd. All rights reserved

Agreement Between Magnetic Resonance Imaging Proton Density Fat Fraction Measurements and Pathologist-assigned Steatosis Grades of Liver Biopsies from Adults with Nonalcoholic Steatohepatitis

Background & Aims We assessed the diagnostic performance of magnetic resonance imaging (MRI) proton density fat fraction (PDFF) in grading hepatic steatosis and change in hepatic steatosis in adults with nonalcoholic steatohepatitis (NASH) in a multi-center study, using central histology as reference. Methods We collected data from 113 adults with NASH participating in a multi-center, randomized, double-masked, placebo-controlled, phase 2b trial to compare the efficacy cross-sectionally and longitudinally of obeticholic acid vs placebo. Hepatic steatosis was assessed at baseline and after 72 weeks of obeticholic acid or placebo by liver biopsy and MRI (scanners from different manufacturers, at 1.5T or 3T). We compared steatosis estimates by PDFF vs histology. Histologic steatosis grade was scored in consensus by a pathology committee. Cross-validated receiver operating characteristic (ROC) analyses were performed. Results At baseline, 34% of subjects had steatosis grade 0 or 1, 39% had steatosis grade 2, and 27% had steatosis grade 3; corresponding mean PDFF values were 9.8%±3.7%, 18.1%±4.3%, and 30.1%±8.1%. PDFF classified steatosis grade 0–1 vs 2–3 with an area under the ROC curve (AUROC) of 0.95 (95% CI, 0.91–0.98), and grade 0–2 vs grade 3 steatosis with an AUROC of 0.96 (95% CI, 0.93–0.99). PDFF cut-off values at 90% specificity were 16.3% for grades 2–3 and 21.7% for grade 3, with corresponding sensitivities of 83% and 84%. After 72 weeks' of obeticholic vs placebo, 42% of subjects had a reduced steatosis grade (mean reduction in PDFF from baseline of 7.4%±8.7%), 49% had no change in steatosis grade (mean increase in PDFF from baseline of 0.3%±6.3%), and 9% had an increased steatosis grade (mean increase in PDFF from baseline of 7.7%±6.0%). PDFF change identified subjects with reduced steatosis grade with an AUROC of 0.81 (95% CI, 0.71–0.91) and increased steatosis grade with an AUROC of 0.81 (95% CI, 0.63–0.99). A PDFF reduction of 5.15% identified subjects with reduced steatosis grade with 90% specificity and 58% sensitivity, whereas a PDFF increase of 5.6% identified those with increased steatosis grade with 90% specificity and 57% sensitivity. Conclusions Based on data from a phase 2 randomized controlled trial of adults with NASH, PDFF estimated by MRI scanners of different field strength and at different sites, accurately classifies grades and changes in hepatic steatosis when histologic analysis of biopsies is used as a reference

Collisional kinetics of non-uniform electric field, low-pressure, direct-current discharges in H2_{2}

A model of the collisional kinetics of energetic hydrogen atoms, molecules, and ions in pure H$_2$ discharges is used to predict H$_\alpha$ emission profiles and spatial distributions of emission from the cathode regions of low-pressure, weakly-ionized discharges for comparison with a wide variety of experiments. Positive and negative ion energy distributions are also predicted. The model developed for spatially uniform electric fields and current densities less than $10^{-3}$ A/m$^2$ is extended to non-uniform electric fields, current densities of $10^{3}$ A/m$^2$, and electric field to gas density ratios $E/N = 1.3$ MTd at 0.002 to 5 Torr pressure. (1 Td = $10^{-21}$ V m$^2$ and 1 Torr = 133 Pa) The observed far-wing Doppler broadening and spatial distribution of the H$_\alpha$ emission is consistent with reactions among H$^+$, H$_2^+$, H$_3^+$, and H$^-H$ ions, fast H atoms, and fast H$_2$ molecules, and with reflection, excitation, and attachment to fast H atoms at surfaces. The H$_\alpha$ excitation and H$^-$ formation occur principally by collisions of fast H, fast H$_2$, and H$^+$ with H$_2$. Simplifications include using a one-dimensional geometry, a multi-beam transport model, and the average cathode-fall electric field. The H$_\alpha$ emission is linear with current density over eight orders of magnitude. The calculated ion energy distributions agree satisfactorily with experiment for H$_2^+$ and H$_3^+$, but are only in qualitative agreement for H$^+$ and H$^-$. The experiments successfully modeled range from short-gap, parallel-plane glow discharges to beam-like, electrostatic-confinement discharges.Comment: Submitted to Plasmas Sources Science and Technology 8/18/201

Direct and Inverse Variational Problems on Time Scales: A Survey

We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to attain a local minimum at a given point of the vector space. Furthermore, we provide a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation (Helmholtz's problem of the calculus of variations on time scales). New and interesting results for the discrete and quantum settings are obtained as particular cases. Finally, we consider very general problems of the calculus of variations given by the composition of a certain scalar function with delta and nabla integrals of a vector valued field.Comment: This is a preprint of a paper whose final and definite form will be published in the Springer Volume 'Modeling, Dynamics, Optimization and Bioeconomics II', Edited by A. A. Pinto and D. Zilberman (Eds.), Springer Proceedings in Mathematics & Statistics. Submitted 03/Sept/2014; Accepted, after a revision, 19/Jan/201